LMFDB - Embedded newform 4028.2.a.d.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(1,4028)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.1637419342$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 30 x^{17} + 132 x^{16} + 332 x^{15} - 1714 x^{14} - 1598 x^{13} + 11179 x^{12} + \cdots - 49 $ x^19 - 4x^18 - 30x^17 + 132x^16 + 332x^15 - 1714x^14 - 1598x^13 + 11179x^12 + 2544x^11 - 38897x^10 + 3416x^9 + 71354x^8 - 10941x^7 - 64854x^6 + 219x^5 + 26515x^4 + 5796x^3 - 2375x^2 - 826x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-1.67288$ of defining polynomial
Character	$\chi$	$=$	4028.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.67288 q^{3} -0.778334 q^{5} +3.24553 q^{7} -0.201474 q^{9} +O(q^{10})$	$q+1.67288 q^{3} -0.778334 q^{5} +3.24553 q^{7} -0.201474 q^{9} -5.19304 q^{11} -4.82763 q^{13} -1.30206 q^{15} +1.25277 q^{17} -1.00000 q^{19} +5.42938 q^{21} +8.02017 q^{23} -4.39420 q^{25} -5.35568 q^{27} -3.73699 q^{29} +8.22083 q^{31} -8.68733 q^{33} -2.52611 q^{35} -4.60720 q^{37} -8.07604 q^{39} -7.33839 q^{41} -7.17686 q^{43} +0.156814 q^{45} +2.69328 q^{47} +3.53345 q^{49} +2.09574 q^{51} +1.00000 q^{53} +4.04192 q^{55} -1.67288 q^{57} +3.81793 q^{59} -2.56764 q^{61} -0.653888 q^{63} +3.75751 q^{65} -6.53327 q^{67} +13.4168 q^{69} +5.85114 q^{71} -11.1689 q^{73} -7.35096 q^{75} -16.8542 q^{77} -2.01851 q^{79} -8.35499 q^{81} -8.37951 q^{83} -0.975075 q^{85} -6.25153 q^{87} -14.3217 q^{89} -15.6682 q^{91} +13.7525 q^{93} +0.778334 q^{95} +8.42831 q^{97} +1.04626 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.67288	0.965838	0.482919	−	0.875665i	$-0.339577\pi$
$3$	1.67288	0.965838	0.482919	+	0.875665i	$0.339577\pi$
$4$	0	0
$5$	−0.778334	−0.348082	−0.174041	−	0.984738i	$-0.555682\pi$
$5$	−0.778334	−0.348082	−0.174041	+	0.984738i	$0.555682\pi$
$6$	0	0
$7$	3.24553	1.22669	0.613347	−	0.789813i	$-0.289823\pi$
$7$	3.24553	1.22669	0.613347	+	0.789813i	$0.289823\pi$
$8$	0	0
$9$	−0.201474	−0.0671579
$10$	0	0
$11$	−5.19304	−1.56576	−0.782880	−	0.622173i	$-0.786250\pi$
$11$	−5.19304	−1.56576	−0.782880	+	0.622173i	$0.786250\pi$
$12$	0	0
$13$	−4.82763	−1.33894	−0.669472	−	0.742838i	$-0.733479\pi$
$13$	−4.82763	−1.33894	−0.669472	+	0.742838i	$0.733479\pi$
$14$	0	0
$15$	−1.30206	−0.336190
$16$	0	0
$17$	1.25277	0.303842	0.151921	−	0.988393i	$-0.451454\pi$
$17$	1.25277	0.303842	0.151921	+	0.988393i	$0.451454\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	5.42938	1.18479
$22$	0	0
$23$	8.02017	1.67232	0.836161	−	0.548484i	$-0.184795\pi$
$23$	8.02017	1.67232	0.836161	+	0.548484i	$0.184795\pi$
$24$	0	0
$25$	−4.39420	−0.878839
$26$	0	0
$27$	−5.35568	−1.03070
$28$	0	0
$29$	−3.73699	−0.693941	−0.346971	−	0.937876i	$-0.612790\pi$
$29$	−3.73699	−0.693941	−0.346971	+	0.937876i	$0.612790\pi$
$30$	0	0
$31$	8.22083	1.47651	0.738253	−	0.674524i	$-0.235651\pi$
$31$	8.22083	1.47651	0.738253	+	0.674524i	$0.235651\pi$
$32$	0	0
$33$	−8.68733	−1.51227
$34$	0	0
$35$	−2.52611	−0.426990
$36$	0	0
$37$	−4.60720	−0.757419	−0.378710	−	0.925516i	$-0.623632\pi$
$37$	−4.60720	−0.757419	−0.378710	+	0.925516i	$0.623632\pi$
$38$	0	0
$39$	−8.07604	−1.29320
$40$	0	0
$41$	−7.33839	−1.14606	−0.573032	−	0.819533i	$-0.694233\pi$
$41$	−7.33839	−1.14606	−0.573032	+	0.819533i	$0.694233\pi$
$42$	0	0
$43$	−7.17686	−1.09446	−0.547230	−	0.836982i	$-0.684318\pi$
$43$	−7.17686	−1.09446	−0.547230	+	0.836982i	$0.684318\pi$
$44$	0	0
$45$	0.156814	0.0233764
$46$	0	0
$47$	2.69328	0.392855	0.196427	−	0.980518i	$-0.437066\pi$
$47$	2.69328	0.392855	0.196427	+	0.980518i	$0.437066\pi$
$48$	0	0
$49$	3.53345	0.504779
$50$	0	0
$51$	2.09574	0.293462
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	4.04192	0.545012
$56$	0	0
$57$	−1.67288	−0.221578
$58$	0	0
$59$	3.81793	0.497053	0.248526	−	0.968625i	$-0.420054\pi$
$59$	3.81793	0.497053	0.248526	+	0.968625i	$0.420054\pi$
$60$	0	0
$61$	−2.56764	−0.328753	−0.164376	−	0.986398i	$-0.552561\pi$
$61$	−2.56764	−0.328753	−0.164376	+	0.986398i	$0.552561\pi$
$62$	0	0
$63$	−0.653888	−0.0823822
$64$	0	0
$65$	3.75751	0.466061
$66$	0	0
$67$	−6.53327	−0.798166	−0.399083	−	0.916915i	$-0.630671\pi$
$67$	−6.53327	−0.798166	−0.399083	+	0.916915i	$0.630671\pi$
$68$	0	0
$69$	13.4168	1.61519
$70$	0	0
$71$	5.85114	0.694403	0.347202	−	0.937791i	$-0.387132\pi$
$71$	5.85114	0.694403	0.347202	+	0.937791i	$0.387132\pi$
$72$	0	0
$73$	−11.1689	−1.30722	−0.653609	−	0.756833i	$-0.726746\pi$
$73$	−11.1689	−1.30722	−0.653609	+	0.756833i	$0.726746\pi$
$74$	0	0
$75$	−7.35096	−0.848816
$76$	0	0
$77$	−16.8542	−1.92071
$78$	0	0
$79$	−2.01851	−0.227100	−0.113550	−	0.993532i	$-0.536222\pi$
$79$	−2.01851	−0.227100	−0.113550	+	0.993532i	$0.536222\pi$
$80$	0	0
$81$	−8.35499	−0.928332
$82$	0	0
$83$	−8.37951	−0.919771	−0.459886	−	0.887978i	$-0.652110\pi$
$83$	−8.37951	−0.919771	−0.459886	+	0.887978i	$0.652110\pi$
$84$	0	0
$85$	−0.975075	−0.105762
$86$	0	0
$87$	−6.25153	−0.670235
$88$	0	0
$89$	−14.3217	−1.51810	−0.759048	−	0.651035i	$-0.774335\pi$
$89$	−14.3217	−1.51810	−0.759048	+	0.651035i	$0.774335\pi$
$90$	0	0
$91$	−15.6682	−1.64247
$92$	0	0
$93$	13.7525	1.42606
$94$	0	0
$95$	0.778334	0.0798554
$96$	0	0
$97$	8.42831	0.855766	0.427883	−	0.903834i	$-0.359260\pi$
$97$	8.42831	0.855766	0.427883	+	0.903834i	$0.359260\pi$
$98$	0	0
$99$	1.04626	0.105153
$100$	0	0
$101$	−5.67029	−0.564215	−0.282108	−	0.959383i	$-0.591034\pi$
$101$	−5.67029	−0.564215	−0.282108	+	0.959383i	$0.591034\pi$
$102$	0	0
$103$	−15.8524	−1.56199	−0.780994	−	0.624539i	$-0.785287\pi$
$103$	−15.8524	−1.56199	−0.780994	+	0.624539i	$0.785287\pi$
$104$	0	0
$105$	−4.22587	−0.412403
$106$	0	0
$107$	−10.7383	−1.03811	−0.519053	−	0.854742i	$-0.673715\pi$
$107$	−10.7383	−1.03811	−0.519053	+	0.854742i	$0.673715\pi$
$108$	0	0
$109$	−14.5132	−1.39011	−0.695055	−	0.718957i	$-0.744620\pi$
$109$	−14.5132	−1.39011	−0.695055	+	0.718957i	$0.744620\pi$
$110$	0	0
$111$	−7.70730	−0.731544
$112$	0	0
$113$	7.07224	0.665301	0.332650	−	0.943050i	$-0.392057\pi$
$113$	7.07224	0.665301	0.332650	+	0.943050i	$0.392057\pi$
$114$	0	0
$115$	−6.24238	−0.582105
$116$	0	0
$117$	0.972640	0.0899206
$118$	0	0
$119$	4.06591	0.372721
$120$	0	0
$121$	15.9677	1.45161
$122$	0	0
$123$	−12.2762	−1.10691
$124$	0	0
$125$	7.31182	0.653989
$126$	0	0
$127$	5.27496	0.468077	0.234039	−	0.972227i	$-0.424806\pi$
$127$	5.27496	0.468077	0.234039	+	0.972227i	$0.424806\pi$
$128$	0	0
$129$	−12.0060	−1.05707
$130$	0	0
$131$	1.82656	0.159587	0.0797934	−	0.996811i	$-0.474574\pi$
$131$	1.82656	0.159587	0.0797934	+	0.996811i	$0.474574\pi$
$132$	0	0
$133$	−3.24553	−0.281423
$134$	0	0
$135$	4.16851	0.358768
$136$	0	0
$137$	−14.5688	−1.24470	−0.622348	−	0.782741i	$-0.713821\pi$
$137$	−14.5688	−1.24470	−0.622348	+	0.782741i	$0.713821\pi$
$138$	0	0
$139$	23.2106	1.96870	0.984348	−	0.176235i	$-0.0563919\pi$
$139$	23.2106	1.96870	0.984348	+	0.176235i	$0.0563919\pi$
$140$	0	0
$141$	4.50553	0.379434
$142$	0	0
$143$	25.0701	2.09646
$144$	0	0
$145$	2.90863	0.241548
$146$	0	0
$147$	5.91104	0.487534
$148$	0	0
$149$	−5.00196	−0.409777	−0.204888	−	0.978785i	$-0.565683\pi$
$149$	−5.00196	−0.409777	−0.204888	+	0.978785i	$0.565683\pi$
$150$	0	0
$151$	15.5101	1.26219	0.631097	−	0.775704i	$-0.282605\pi$
$151$	15.5101	1.26219	0.631097	+	0.775704i	$0.282605\pi$
$152$	0	0
$153$	−0.252400	−0.0204054
$154$	0	0
$155$	−6.39855	−0.513944
$156$	0	0
$157$	−19.4443	−1.55183	−0.775913	−	0.630840i	$-0.782710\pi$
$157$	−19.4443	−1.55183	−0.775913	+	0.630840i	$0.782710\pi$
$158$	0	0
$159$	1.67288	0.132668
$160$	0	0
$161$	26.0297	2.05143
$162$	0	0
$163$	11.9070	0.932630	0.466315	−	0.884619i	$-0.345581\pi$
$163$	11.9070	0.932630	0.466315	+	0.884619i	$0.345581\pi$
$164$	0	0
$165$	6.76165	0.526393
$166$	0	0
$167$	24.3850	1.88697	0.943485	−	0.331414i	$-0.107526\pi$
$167$	24.3850	1.88697	0.943485	+	0.331414i	$0.107526\pi$
$168$	0	0
$169$	10.3060	0.792768
$170$	0	0
$171$	0.201474	0.0154071
$172$	0	0
$173$	22.9883	1.74777	0.873883	−	0.486137i	$-0.161594\pi$
$173$	22.9883	1.74777	0.873883	+	0.486137i	$0.161594\pi$
$174$	0	0
$175$	−14.2615	−1.07807
$176$	0	0
$177$	6.38694	0.480072
$178$	0	0
$179$	11.0530	0.826141	0.413070	−	0.910699i	$-0.364456\pi$
$179$	11.0530	0.826141	0.413070	+	0.910699i	$0.364456\pi$
$180$	0	0
$181$	−20.7089	−1.53928	−0.769640	−	0.638478i	$-0.779564\pi$
$181$	−20.7089	−1.53928	−0.769640	+	0.638478i	$0.779564\pi$
$182$	0	0
$183$	−4.29535	−0.317522
$184$	0	0
$185$	3.58594	0.263644
$186$	0	0
$187$	−6.50569	−0.475743
$188$	0	0
$189$	−17.3820	−1.26436
$190$	0	0
$191$	−1.06309	−0.0769226	−0.0384613	−	0.999260i	$-0.512246\pi$
$191$	−1.06309	−0.0769226	−0.0384613	+	0.999260i	$0.512246\pi$
$192$	0	0
$193$	−11.2404	−0.809098	−0.404549	−	0.914516i	$-0.632572\pi$
$193$	−11.2404	−0.809098	−0.404549	+	0.914516i	$0.632572\pi$
$194$	0	0
$195$	6.28586	0.450140
$196$	0	0
$197$	−10.5120	−0.748952	−0.374476	−	0.927237i	$-0.622177\pi$
$197$	−10.5120	−0.748952	−0.374476	+	0.927237i	$0.622177\pi$
$198$	0	0
$199$	−6.37118	−0.451641	−0.225821	−	0.974169i	$-0.572506\pi$
$199$	−6.37118	−0.451641	−0.225821	+	0.974169i	$0.572506\pi$
$200$	0	0
$201$	−10.9294	−0.770898
$202$	0	0
$203$	−12.1285	−0.851254
$204$	0	0
$205$	5.71172	0.398924
$206$	0	0
$207$	−1.61585	−0.112310
$208$	0	0
$209$	5.19304	0.359210
$210$	0	0
$211$	−0.705035	−0.0485366	−0.0242683	−	0.999705i	$-0.507726\pi$
$211$	−0.705035	−0.0485366	−0.0242683	+	0.999705i	$0.507726\pi$
$212$	0	0
$213$	9.78826	0.670681
$214$	0	0
$215$	5.58599	0.380961
$216$	0	0
$217$	26.6809	1.81122
$218$	0	0
$219$	−18.6842	−1.26256
$220$	0	0
$221$	−6.04791	−0.406827
$222$	0	0
$223$	13.3195	0.891940	0.445970	−	0.895048i	$-0.352859\pi$
$223$	13.3195	0.891940	0.445970	+	0.895048i	$0.352859\pi$
$224$	0	0
$225$	0.885315	0.0590210
$226$	0	0
$227$	2.89726	0.192298	0.0961490	−	0.995367i	$-0.469347\pi$
$227$	2.89726	0.192298	0.0961490	+	0.995367i	$0.469347\pi$
$228$	0	0
$229$	−13.3336	−0.881111	−0.440556	−	0.897725i	$-0.645219\pi$
$229$	−13.3336	−0.881111	−0.440556	+	0.897725i	$0.645219\pi$
$230$	0	0
$231$	−28.1950	−1.85509
$232$	0	0
$233$	12.8323	0.840671	0.420335	−	0.907369i	$-0.361912\pi$
$233$	12.8323	0.840671	0.420335	+	0.907369i	$0.361912\pi$
$234$	0	0
$235$	−2.09627	−0.136745
$236$	0	0
$237$	−3.37672	−0.219342
$238$	0	0
$239$	1.49143	0.0964724	0.0482362	−	0.998836i	$-0.484640\pi$
$239$	1.49143	0.0964724	0.0482362	+	0.998836i	$0.484640\pi$
$240$	0	0
$241$	5.34389	0.344230	0.172115	−	0.985077i	$-0.444940\pi$
$241$	5.34389	0.344230	0.172115	+	0.985077i	$0.444940\pi$
$242$	0	0
$243$	2.09015	0.134083
$244$	0	0
$245$	−2.75021	−0.175704
$246$	0	0
$247$	4.82763	0.307175
$248$	0	0
$249$	−14.0179	−0.888349
$250$	0	0
$251$	−19.1204	−1.20687	−0.603436	−	0.797411i	$-0.706202\pi$
$251$	−19.1204	−1.20687	−0.603436	+	0.797411i	$0.706202\pi$
$252$	0	0
$253$	−41.6491	−2.61846
$254$	0	0
$255$	−1.63118	−0.102149
$256$	0	0
$257$	−12.5390	−0.782161	−0.391080	−	0.920357i	$-0.627899\pi$
$257$	−12.5390	−0.782161	−0.391080	+	0.920357i	$0.627899\pi$
$258$	0	0
$259$	−14.9528	−0.929122
$260$	0	0
$261$	0.752905	0.0466036
$262$	0	0
$263$	15.6258	0.963530	0.481765	−	0.876300i	$-0.339996\pi$
$263$	15.6258	0.963530	0.481765	+	0.876300i	$0.339996\pi$
$264$	0	0
$265$	−0.778334	−0.0478127
$266$	0	0
$267$	−23.9585	−1.46623
$268$	0	0
$269$	25.4952	1.55447	0.777235	−	0.629211i	$-0.216622\pi$
$269$	25.4952	1.55447	0.777235	+	0.629211i	$0.216622\pi$
$270$	0	0
$271$	13.9737	0.848840	0.424420	−	0.905466i	$-0.360478\pi$
$271$	13.9737	0.848840	0.424420	+	0.905466i	$0.360478\pi$
$272$	0	0
$273$	−26.2110	−1.58636
$274$	0	0
$275$	22.8192	1.37605
$276$	0	0
$277$	1.83060	0.109990	0.0549950	−	0.998487i	$-0.482486\pi$
$277$	1.83060	0.109990	0.0549950	+	0.998487i	$0.482486\pi$
$278$	0	0
$279$	−1.65628	−0.0991589
$280$	0	0
$281$	5.13605	0.306391	0.153195	−	0.988196i	$-0.451044\pi$
$281$	5.13605	0.306391	0.153195	+	0.988196i	$0.451044\pi$
$282$	0	0
$283$	29.2044	1.73602	0.868010	−	0.496547i	$-0.165399\pi$
$283$	29.2044	1.73602	0.868010	+	0.496547i	$0.165399\pi$
$284$	0	0
$285$	1.30206	0.0771273
$286$	0	0
$287$	−23.8169	−1.40587
$288$	0	0
$289$	−15.4306	−0.907680
$290$	0	0
$291$	14.0996	0.826531
$292$	0	0
$293$	29.1212	1.70128	0.850638	−	0.525752i	$-0.176216\pi$
$293$	29.1212	1.70128	0.850638	+	0.525752i	$0.176216\pi$
$294$	0	0
$295$	−2.97163	−0.173015
$296$	0	0
$297$	27.8123	1.61383
$298$	0	0
$299$	−38.7184	−2.23914
$300$	0	0
$301$	−23.2927	−1.34257
$302$	0	0
$303$	−9.48572	−0.544940
$304$	0	0
$305$	1.99848	0.114433
$306$	0	0
$307$	2.49151	0.142198	0.0710991	−	0.997469i	$-0.477349\pi$
$307$	2.49151	0.142198	0.0710991	+	0.997469i	$0.477349\pi$
$308$	0	0
$309$	−26.5192	−1.50863
$310$	0	0
$311$	−2.81819	−0.159805	−0.0799025	−	0.996803i	$-0.525461\pi$
$311$	−2.81819	−0.159805	−0.0799025	+	0.996803i	$0.525461\pi$
$312$	0	0
$313$	−33.3226	−1.88350	−0.941752	−	0.336309i	$-0.890821\pi$
$313$	−33.3226	−1.88350	−0.941752	+	0.336309i	$0.890821\pi$
$314$	0	0
$315$	0.508944	0.0286757
$316$	0	0
$317$	−26.8703	−1.50919	−0.754593	−	0.656193i	$-0.772166\pi$
$317$	−26.8703	−1.50919	−0.754593	+	0.656193i	$0.772166\pi$
$318$	0	0
$319$	19.4063	1.08655
$320$	0	0
$321$	−17.9638	−1.00264
$322$	0	0
$323$	−1.25277	−0.0697061
$324$	0	0
$325$	21.2135	1.17672
$326$	0	0
$327$	−24.2788	−1.34262
$328$	0	0
$329$	8.74110	0.481913
$330$	0	0
$331$	0.719643	0.0395552	0.0197776	−	0.999804i	$-0.493704\pi$
$331$	0.719643	0.0395552	0.0197776	+	0.999804i	$0.493704\pi$
$332$	0	0
$333$	0.928230	0.0508667
$334$	0	0
$335$	5.08507	0.277827
$336$	0	0
$337$	−1.71348	−0.0933392	−0.0466696	−	0.998910i	$-0.514861\pi$
$337$	−1.71348	−0.0933392	−0.0466696	+	0.998910i	$0.514861\pi$
$338$	0	0
$339$	11.8310	0.642572
$340$	0	0
$341$	−42.6911	−2.31185
$342$	0	0
$343$	−11.2508	−0.607485
$344$	0	0
$345$	−10.4427	−0.562218
$346$	0	0
$347$	19.6894	1.05698	0.528490	−	0.848940i	$-0.322758\pi$
$347$	19.6894	1.05698	0.528490	+	0.848940i	$0.322758\pi$
$348$	0	0
$349$	−26.6881	−1.42858	−0.714291	−	0.699849i	$-0.753250\pi$
$349$	−26.6881	−1.42858	−0.714291	+	0.699849i	$0.753250\pi$
$350$	0	0
$351$	25.8552	1.38005
$352$	0	0
$353$	25.2447	1.34364	0.671819	−	0.740716i	$-0.265513\pi$
$353$	25.2447	1.34364	0.671819	+	0.740716i	$0.265513\pi$
$354$	0	0
$355$	−4.55415	−0.241709
$356$	0	0
$357$	6.80177	0.359988
$358$	0	0
$359$	6.47017	0.341482	0.170741	−	0.985316i	$-0.445384\pi$
$359$	6.47017	0.341482	0.170741	+	0.985316i	$0.445384\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	26.7120	1.40201
$364$	0	0
$365$	8.69312	0.455018
$366$	0	0
$367$	−16.1206	−0.841487	−0.420744	−	0.907180i	$-0.638231\pi$
$367$	−16.1206	−0.841487	−0.420744	+	0.907180i	$0.638231\pi$
$368$	0	0
$369$	1.47849	0.0769672
$370$	0	0
$371$	3.24553	0.168499
$372$	0	0
$373$	31.3368	1.62256	0.811278	−	0.584660i	$-0.198772\pi$
$373$	31.3368	1.62256	0.811278	+	0.584660i	$0.198772\pi$
$374$	0	0
$375$	12.2318	0.631647
$376$	0	0
$377$	18.0408	0.929148
$378$	0	0
$379$	−17.2986	−0.888571	−0.444286	−	0.895885i	$-0.646542\pi$
$379$	−17.2986	−0.888571	−0.444286	+	0.895885i	$0.646542\pi$
$380$	0	0
$381$	8.82437	0.452086
$382$	0	0
$383$	−30.4824	−1.55758	−0.778789	−	0.627286i	$-0.784166\pi$
$383$	−30.4824	−1.55758	−0.778789	+	0.627286i	$0.784166\pi$
$384$	0	0
$385$	13.1182	0.668564
$386$	0	0
$387$	1.44595	0.0735016
$388$	0	0
$389$	17.3858	0.881495	0.440747	−	0.897631i	$-0.354713\pi$
$389$	17.3858	0.881495	0.440747	+	0.897631i	$0.354713\pi$
$390$	0	0
$391$	10.0474	0.508121
$392$	0	0
$393$	3.05561	0.154135
$394$	0	0
$395$	1.57107	0.0790494
$396$	0	0
$397$	−26.2010	−1.31499	−0.657496	−	0.753458i	$-0.728384\pi$
$397$	−26.2010	−1.31499	−0.657496	+	0.753458i	$0.728384\pi$
$398$	0	0
$399$	−5.42938	−0.271809
$400$	0	0
$401$	24.2367	1.21032	0.605161	−	0.796103i	$-0.293109\pi$
$401$	24.2367	1.21032	0.605161	+	0.796103i	$0.293109\pi$
$402$	0	0
$403$	−39.6871	−1.97696
$404$	0	0
$405$	6.50297	0.323135
$406$	0	0
$407$	23.9254	1.18594
$408$	0	0
$409$	−26.9488	−1.33253	−0.666266	−	0.745714i	$-0.732108\pi$
$409$	−26.9488	−1.33253	−0.666266	+	0.745714i	$0.732108\pi$
$410$	0	0
$411$	−24.3718	−1.20217
$412$	0	0
$413$	12.3912	0.609732
$414$	0	0
$415$	6.52206	0.320155
$416$	0	0
$417$	38.8285	1.90144
$418$	0	0
$419$	−16.0627	−0.784714	−0.392357	−	0.919813i	$-0.628340\pi$
$419$	−16.0627	−0.784714	−0.392357	+	0.919813i	$0.628340\pi$
$420$	0	0
$421$	−10.9619	−0.534250	−0.267125	−	0.963662i	$-0.586074\pi$
$421$	−10.9619	−0.534250	−0.267125	+	0.963662i	$0.586074\pi$
$422$	0	0
$423$	−0.542624	−0.0263833
$424$	0	0
$425$	−5.50492	−0.267028
$426$	0	0
$427$	−8.33335	−0.403279
$428$	0	0
$429$	41.9392	2.02484
$430$	0	0
$431$	−14.3717	−0.692260	−0.346130	−	0.938187i	$-0.612504\pi$
$431$	−14.3717	−0.692260	−0.346130	+	0.938187i	$0.612504\pi$
$432$	0	0
$433$	−10.0544	−0.483184	−0.241592	−	0.970378i	$-0.577670\pi$
$433$	−10.0544	−0.483184	−0.241592	+	0.970378i	$0.577670\pi$
$434$	0	0
$435$	4.86578	0.233296
$436$	0	0
$437$	−8.02017	−0.383657
$438$	0	0
$439$	−6.33631	−0.302416	−0.151208	−	0.988502i	$-0.548316\pi$
$439$	−6.33631	−0.302416	−0.151208	+	0.988502i	$0.548316\pi$
$440$	0	0
$441$	−0.711897	−0.0338999
$442$	0	0
$443$	−4.84937	−0.230401	−0.115200	−	0.993342i	$-0.536751\pi$
$443$	−4.84937	−0.230401	−0.115200	+	0.993342i	$0.536751\pi$
$444$	0	0
$445$	11.1471	0.528421
$446$	0	0
$447$	−8.36768	−0.395778
$448$	0	0
$449$	24.0916	1.13695	0.568476	−	0.822700i	$-0.307533\pi$
$449$	24.0916	1.13695	0.568476	+	0.822700i	$0.307533\pi$
$450$	0	0
$451$	38.1085	1.79446
$452$	0	0
$453$	25.9465	1.21907
$454$	0	0
$455$	12.1951	0.571715
$456$	0	0
$457$	21.3811	1.00017	0.500084	−	0.865977i	$-0.333302\pi$
$457$	21.3811	1.00017	0.500084	+	0.865977i	$0.333302\pi$
$458$	0	0
$459$	−6.70944	−0.313170
$460$	0	0
$461$	14.1902	0.660902	0.330451	−	0.943823i	$-0.392799\pi$
$461$	14.1902	0.660902	0.330451	+	0.943823i	$0.392799\pi$
$462$	0	0
$463$	−13.1047	−0.609029	−0.304514	−	0.952508i	$-0.598494\pi$
$463$	−13.1047	−0.609029	−0.304514	+	0.952508i	$0.598494\pi$
$464$	0	0
$465$	−10.7040	−0.496387
$466$	0	0
$467$	−6.78909	−0.314161	−0.157081	−	0.987586i	$-0.550208\pi$
$467$	−6.78909	−0.314161	−0.157081	+	0.987586i	$0.550208\pi$
$468$	0	0
$469$	−21.2039	−0.979105
$470$	0	0
$471$	−32.5280	−1.49881
$472$	0	0
$473$	37.2697	1.71366
$474$	0	0
$475$	4.39420	0.201620
$476$	0	0
$477$	−0.201474	−0.00922484
$478$	0	0
$479$	−22.5334	−1.02958	−0.514789	−	0.857317i	$-0.672130\pi$
$479$	−22.5334	−1.02958	−0.514789	+	0.857317i	$0.672130\pi$
$480$	0	0
$481$	22.2419	1.01414
$482$	0	0
$483$	43.5446	1.98135
$484$	0	0
$485$	−6.56004	−0.297876
$486$	0	0
$487$	10.7864	0.488780	0.244390	−	0.969677i	$-0.421412\pi$
$487$	10.7864	0.488780	0.244390	+	0.969677i	$0.421412\pi$
$488$	0	0
$489$	19.9190	0.900769
$490$	0	0
$491$	32.0432	1.44609	0.723045	−	0.690801i	$-0.242742\pi$
$491$	32.0432	1.44609	0.723045	+	0.690801i	$0.242742\pi$
$492$	0	0
$493$	−4.68159	−0.210848
$494$	0	0
$495$	−0.814340	−0.0366019
$496$	0	0
$497$	18.9901	0.851820
$498$	0	0
$499$	−25.1862	−1.12749	−0.563743	−	0.825950i	$-0.690639\pi$
$499$	−25.1862	−1.12749	−0.563743	+	0.825950i	$0.690639\pi$
$500$	0	0
$501$	40.7932	1.82251
$502$	0	0
$503$	15.7313	0.701426	0.350713	−	0.936483i	$-0.385939\pi$
$503$	15.7313	0.701426	0.350713	+	0.936483i	$0.385939\pi$
$504$	0	0
$505$	4.41338	0.196393
$506$	0	0
$507$	17.2407	0.765685
$508$	0	0
$509$	26.0165	1.15316	0.576581	−	0.817040i	$-0.304386\pi$
$509$	26.0165	1.15316	0.576581	+	0.817040i	$0.304386\pi$
$510$	0	0
$511$	−36.2489	−1.60356
$512$	0	0
$513$	5.35568	0.236459
$514$	0	0
$515$	12.3385	0.543699
$516$	0	0
$517$	−13.9863	−0.615116
$518$	0	0
$519$	38.4566	1.68806
$520$	0	0
$521$	−29.0583	−1.27307	−0.636533	−	0.771250i	$-0.719632\pi$
$521$	−29.0583	−1.27307	−0.636533	+	0.771250i	$0.719632\pi$
$522$	0	0
$523$	−10.4208	−0.455668	−0.227834	−	0.973700i	$-0.573164\pi$
$523$	−10.4208	−0.455668	−0.227834	+	0.973700i	$0.573164\pi$
$524$	0	0
$525$	−23.8578	−1.04124
$526$	0	0
$527$	10.2988	0.448624
$528$	0	0
$529$	41.3232	1.79666
$530$	0	0
$531$	−0.769213	−0.0333810
$532$	0	0
$533$	35.4270	1.53451
$534$	0	0
$535$	8.35795	0.361346
$536$	0	0
$537$	18.4904	0.797918
$538$	0	0
$539$	−18.3494	−0.790363
$540$	0	0
$541$	−17.4953	−0.752183	−0.376091	−	0.926583i	$-0.622732\pi$
$541$	−17.4953	−0.752183	−0.376091	+	0.926583i	$0.622732\pi$
$542$	0	0
$543$	−34.6435	−1.48669
$544$	0	0
$545$	11.2961	0.483872
$546$	0	0
$547$	1.69061	0.0722852	0.0361426	−	0.999347i	$-0.488493\pi$
$547$	1.69061	0.0722852	0.0361426	+	0.999347i	$0.488493\pi$
$548$	0	0
$549$	0.517312	0.0220783
$550$	0	0
$551$	3.73699	0.159201
$552$	0	0
$553$	−6.55113	−0.278582
$554$	0	0
$555$	5.99885	0.254637
$556$	0	0
$557$	7.27559	0.308277	0.154138	−	0.988049i	$-0.450740\pi$
$557$	7.27559	0.308277	0.154138	+	0.988049i	$0.450740\pi$
$558$	0	0
$559$	34.6472	1.46542
$560$	0	0
$561$	−10.8832	−0.459491
$562$	0	0
$563$	−23.0643	−0.972046	−0.486023	−	0.873946i	$-0.661553\pi$
$563$	−23.0643	−0.972046	−0.486023	+	0.873946i	$0.661553\pi$
$564$	0	0
$565$	−5.50457	−0.231579
$566$	0	0
$567$	−27.1163	−1.13878
$568$	0	0
$569$	9.16968	0.384413	0.192207	−	0.981355i	$-0.438436\pi$
$569$	9.16968	0.384413	0.192207	+	0.981355i	$0.438436\pi$
$570$	0	0
$571$	−11.8650	−0.496535	−0.248268	−	0.968692i	$-0.579861\pi$
$571$	−11.8650	−0.496535	−0.248268	+	0.968692i	$0.579861\pi$
$572$	0	0
$573$	−1.77842	−0.0742947
$574$	0	0
$575$	−35.2422	−1.46970
$576$	0	0
$577$	11.6802	0.486252	0.243126	−	0.969995i	$-0.421827\pi$
$577$	11.6802	0.486252	0.243126	+	0.969995i	$0.421827\pi$
$578$	0	0
$579$	−18.8038	−0.781457
$580$	0	0
$581$	−27.1959	−1.12828
$582$	0	0
$583$	−5.19304	−0.215074
$584$	0	0
$585$	−0.757039	−0.0312997
$586$	0	0
$587$	38.8747	1.60453	0.802266	−	0.596967i	$-0.203628\pi$
$587$	38.8747	1.60453	0.802266	+	0.596967i	$0.203628\pi$
$588$	0	0
$589$	−8.22083	−0.338734
$590$	0	0
$591$	−17.5854	−0.723366
$592$	0	0
$593$	44.9056	1.84405	0.922025	−	0.387129i	$-0.126533\pi$
$593$	44.9056	1.84405	0.922025	+	0.387129i	$0.126533\pi$
$594$	0	0
$595$	−3.16463	−0.129737
$596$	0	0
$597$	−10.6582	−0.436212
$598$	0	0
$599$	14.3121	0.584778	0.292389	−	0.956299i	$-0.405550\pi$
$599$	14.3121	0.584778	0.292389	+	0.956299i	$0.405550\pi$
$600$	0	0
$601$	−17.3112	−0.706140	−0.353070	−	0.935597i	$-0.614862\pi$
$601$	−17.3112	−0.706140	−0.353070	+	0.935597i	$0.614862\pi$
$602$	0	0
$603$	1.31628	0.0536031
$604$	0	0
$605$	−12.4282	−0.505277
$606$	0	0
$607$	−1.16252	−0.0471852	−0.0235926	−	0.999722i	$-0.507510\pi$
$607$	−1.16252	−0.0471852	−0.0235926	+	0.999722i	$0.507510\pi$
$608$	0	0
$609$	−20.2895	−0.822173
$610$	0	0
$611$	−13.0021	−0.526010
$612$	0	0
$613$	0.545581	0.0220358	0.0110179	−	0.999939i	$-0.496493\pi$
$613$	0.545581	0.0220358	0.0110179	+	0.999939i	$0.496493\pi$
$614$	0	0
$615$	9.55502	0.385295
$616$	0	0
$617$	15.1728	0.610834	0.305417	−	0.952219i	$-0.401204\pi$
$617$	15.1728	0.610834	0.305417	+	0.952219i	$0.401204\pi$
$618$	0	0
$619$	−19.8771	−0.798930	−0.399465	−	0.916749i	$-0.630804\pi$
$619$	−19.8771	−0.798930	−0.399465	+	0.916749i	$0.630804\pi$
$620$	0	0
$621$	−42.9535	−1.72366
$622$	0	0
$623$	−46.4814	−1.86224
$624$	0	0
$625$	16.2799	0.651198
$626$	0	0
$627$	8.68733	0.346939
$628$	0	0
$629$	−5.77177	−0.230136
$630$	0	0
$631$	−45.5141	−1.81189	−0.905945	−	0.423396i	$-0.860838\pi$
$631$	−45.5141	−1.81189	−0.905945	+	0.423396i	$0.860838\pi$
$632$	0	0
$633$	−1.17944	−0.0468785
$634$	0	0
$635$	−4.10568	−0.162929
$636$	0	0
$637$	−17.0582	−0.675870
$638$	0	0
$639$	−1.17885	−0.0466346
$640$	0	0
$641$	−43.8459	−1.73181	−0.865904	−	0.500210i	$-0.833256\pi$
$641$	−43.8459	−1.73181	−0.865904	+	0.500210i	$0.833256\pi$
$642$	0	0
$643$	35.2995	1.39208	0.696038	−	0.718005i	$-0.254945\pi$
$643$	35.2995	1.39208	0.696038	+	0.718005i	$0.254945\pi$
$644$	0	0
$645$	9.34469	0.367947
$646$	0	0
$647$	−9.74852	−0.383254	−0.191627	−	0.981468i	$-0.561376\pi$
$647$	−9.74852	−0.383254	−0.191627	+	0.981468i	$0.561376\pi$
$648$	0	0
$649$	−19.8267	−0.778265
$650$	0	0
$651$	44.6340	1.74934
$652$	0	0
$653$	−18.6298	−0.729039	−0.364519	−	0.931196i	$-0.618767\pi$
$653$	−18.6298	−0.729039	−0.364519	+	0.931196i	$0.618767\pi$
$654$	0	0
$655$	−1.42167	−0.0555493
$656$	0	0
$657$	2.25023	0.0877900
$658$	0	0
$659$	−28.5798	−1.11331	−0.556655	−	0.830744i	$-0.687915\pi$
$659$	−28.5798	−1.11331	−0.556655	+	0.830744i	$0.687915\pi$
$660$	0	0
$661$	48.3257	1.87965	0.939827	−	0.341652i	$-0.110986\pi$
$661$	48.3257	1.87965	0.939827	+	0.341652i	$0.110986\pi$
$662$	0	0
$663$	−10.1174	−0.392929
$664$	0	0
$665$	2.52611	0.0979582
$666$	0	0
$667$	−29.9713	−1.16049
$668$	0	0
$669$	22.2819	0.861469
$670$	0	0
$671$	13.3339	0.514748
$672$	0	0
$673$	33.0602	1.27438	0.637189	−	0.770707i	$-0.280097\pi$
$673$	33.0602	1.27438	0.637189	+	0.770707i	$0.280097\pi$
$674$	0	0
$675$	23.5339	0.905821
$676$	0	0
$677$	37.7945	1.45256	0.726280	−	0.687399i	$-0.241247\pi$
$677$	37.7945	1.45256	0.726280	+	0.687399i	$0.241247\pi$
$678$	0	0
$679$	27.3543	1.04976
$680$	0	0
$681$	4.84677	0.185729
$682$	0	0
$683$	4.63831	0.177480	0.0887400	−	0.996055i	$-0.471716\pi$
$683$	4.63831	0.177480	0.0887400	+	0.996055i	$0.471716\pi$
$684$	0	0
$685$	11.3394	0.433256
$686$	0	0
$687$	−22.3056	−0.851010
$688$	0	0
$689$	−4.82763	−0.183918
$690$	0	0
$691$	29.2402	1.11235	0.556175	−	0.831065i	$-0.312268\pi$
$691$	29.2402	1.11235	0.556175	+	0.831065i	$0.312268\pi$
$692$	0	0
$693$	3.39567	0.128991
$694$	0	0
$695$	−18.0656	−0.685267
$696$	0	0
$697$	−9.19332	−0.348222
$698$	0	0
$699$	21.4669	0.811951
$700$	0	0
$701$	−9.20118	−0.347524	−0.173762	−	0.984788i	$-0.555592\pi$
$701$	−9.20118	−0.347524	−0.173762	+	0.984788i	$0.555592\pi$
$702$	0	0
$703$	4.60720	0.173764
$704$	0	0
$705$	−3.50681	−0.132074
$706$	0	0
$707$	−18.4031	−0.692120
$708$	0	0
$709$	52.7394	1.98067	0.990336	−	0.138692i	$-0.0442899\pi$
$709$	52.7394	1.98067	0.990336	+	0.138692i	$0.0442899\pi$
$710$	0	0
$711$	0.406676	0.0152516
$712$	0	0
$713$	65.9325	2.46919
$714$	0	0
$715$	−19.5129	−0.729740
$716$	0	0
$717$	2.49498	0.0931767
$718$	0	0
$719$	25.6194	0.955441	0.477720	−	0.878512i	$-0.341463\pi$
$719$	25.6194	0.955441	0.477720	+	0.878512i	$0.341463\pi$
$720$	0	0
$721$	−51.4495	−1.91608
$722$	0	0
$723$	8.93968	0.332470
$724$	0	0
$725$	16.4211	0.609863
$726$	0	0
$727$	11.9483	0.443139	0.221569	−	0.975145i	$-0.428882\pi$
$727$	11.9483	0.443139	0.221569	+	0.975145i	$0.428882\pi$
$728$	0	0
$729$	28.5615	1.05783
$730$	0	0
$731$	−8.99096	−0.332543
$732$	0	0
$733$	−19.4962	−0.720108	−0.360054	−	0.932932i	$-0.617242\pi$
$733$	−19.4962	−0.720108	−0.360054	+	0.932932i	$0.617242\pi$
$734$	0	0
$735$	−4.60077	−0.169702
$736$	0	0
$737$	33.9275	1.24974
$738$	0	0
$739$	−22.5589	−0.829843	−0.414922	−	0.909857i	$-0.636191\pi$
$739$	−22.5589	−0.829843	−0.414922	+	0.909857i	$0.636191\pi$
$740$	0	0
$741$	8.07604	0.296681
$742$	0	0
$743$	−51.4567	−1.88776	−0.943882	−	0.330282i	$-0.892856\pi$
$743$	−51.4567	−1.88776	−0.943882	+	0.330282i	$0.892856\pi$
$744$	0	0
$745$	3.89320	0.142636
$746$	0	0
$747$	1.68825	0.0617699
$748$	0	0
$749$	−34.8513	−1.27344
$750$	0	0
$751$	43.6486	1.59276	0.796380	−	0.604796i	$-0.206746\pi$
$751$	43.6486	1.59276	0.796380	+	0.604796i	$0.206746\pi$
$752$	0	0
$753$	−31.9862	−1.16564
$754$	0	0
$755$	−12.0720	−0.439347
$756$	0	0
$757$	40.6663	1.47804	0.739021	−	0.673683i	$-0.235289\pi$
$757$	40.6663	1.47804	0.739021	+	0.673683i	$0.235289\pi$
$758$	0	0
$759$	−69.6739	−2.52900
$760$	0	0
$761$	−27.2693	−0.988511	−0.494256	−	0.869317i	$-0.664559\pi$
$761$	−27.2693	−0.988511	−0.494256	+	0.869317i	$0.664559\pi$
$762$	0	0
$763$	−47.1029	−1.70524
$764$	0	0
$765$	0.196452	0.00710273
$766$	0	0
$767$	−18.4316	−0.665525
$768$	0	0
$769$	10.2072	0.368081	0.184041	−	0.982919i	$-0.441082\pi$
$769$	10.2072	0.368081	0.184041	+	0.982919i	$0.441082\pi$
$770$	0	0
$771$	−20.9762	−0.755440
$772$	0	0
$773$	29.1663	1.04904	0.524520	−	0.851398i	$-0.324245\pi$
$773$	29.1663	1.04904	0.524520	+	0.851398i	$0.324245\pi$
$774$	0	0
$775$	−36.1239	−1.29761
$776$	0	0
$777$	−25.0142	−0.897381
$778$	0	0
$779$	7.33839	0.262925
$780$	0	0
$781$	−30.3852	−1.08727
$782$	0	0
$783$	20.0141	0.715246
$784$	0	0
$785$	15.1342	0.540162
$786$	0	0
$787$	2.45265	0.0874275	0.0437138	−	0.999044i	$-0.486081\pi$
$787$	2.45265	0.0874275	0.0437138	+	0.999044i	$0.486081\pi$
$788$	0	0
$789$	26.1401	0.930613
$790$	0	0
$791$	22.9532	0.816121
$792$	0	0
$793$	12.3956	0.440181
$794$	0	0
$795$	−1.30206	−0.0461793
$796$	0	0
$797$	−31.7954	−1.12625	−0.563125	−	0.826371i	$-0.690401\pi$
$797$	−31.7954	−1.12625	−0.563125	+	0.826371i	$0.690401\pi$
$798$	0	0
$799$	3.37406	0.119366
$800$	0	0
$801$	2.88544	0.101952
$802$	0	0
$803$	58.0004	2.04679
$804$	0	0
$805$	−20.2598	−0.714064
$806$	0	0
$807$	42.6504	1.50137
$808$	0	0
$809$	−4.20199	−0.147734	−0.0738671	−	0.997268i	$-0.523534\pi$
$809$	−4.20199	−0.147734	−0.0738671	+	0.997268i	$0.523534\pi$
$810$	0	0
$811$	1.63821	0.0575252	0.0287626	−	0.999586i	$-0.490843\pi$
$811$	1.63821	0.0575252	0.0287626	+	0.999586i	$0.490843\pi$
$812$	0	0
$813$	23.3763	0.819841
$814$	0	0
$815$	−9.26764	−0.324631
$816$	0	0
$817$	7.17686	0.251086
$818$	0	0
$819$	3.15673	0.110305
$820$	0	0
$821$	−49.7738	−1.73712	−0.868559	−	0.495585i	$-0.834954\pi$
$821$	−49.7738	−1.73712	−0.868559	+	0.495585i	$0.834954\pi$
$822$	0	0
$823$	−39.2522	−1.36825	−0.684123	−	0.729367i	$-0.739815\pi$
$823$	−39.2522	−1.36825	−0.684123	+	0.729367i	$0.739815\pi$
$824$	0	0
$825$	38.1738	1.32904
$826$	0	0
$827$	13.0806	0.454858	0.227429	−	0.973795i	$-0.426968\pi$
$827$	13.0806	0.454858	0.227429	+	0.973795i	$0.426968\pi$
$828$	0	0
$829$	−16.9126	−0.587398	−0.293699	−	0.955898i	$-0.594886\pi$
$829$	−16.9126	−0.587398	−0.293699	+	0.955898i	$0.594886\pi$
$830$	0	0
$831$	3.06237	0.106233
$832$	0	0
$833$	4.42661	0.153373
$834$	0	0
$835$	−18.9797	−0.656820
$836$	0	0
$837$	−44.0282	−1.52184
$838$	0	0
$839$	22.8199	0.787830	0.393915	−	0.919147i	$-0.371120\pi$
$839$	22.8199	0.787830	0.393915	+	0.919147i	$0.371120\pi$
$840$	0	0
$841$	−15.0349	−0.518445
$842$	0	0
$843$	8.59199	0.295924
$844$	0	0
$845$	−8.02150	−0.275948
$846$	0	0
$847$	51.8235	1.78068
$848$	0	0
$849$	48.8554	1.67671
$850$	0	0
$851$	−36.9506	−1.26665
$852$	0	0
$853$	−42.3169	−1.44890	−0.724451	−	0.689327i	$-0.757907\pi$
$853$	−42.3169	−1.44890	−0.724451	+	0.689327i	$0.757907\pi$
$854$	0	0
$855$	−0.156814	−0.00536292
$856$	0	0
$857$	40.4513	1.38179	0.690895	−	0.722955i	$-0.257217\pi$
$857$	40.4513	1.38179	0.690895	+	0.722955i	$0.257217\pi$
$858$	0	0
$859$	33.4670	1.14188	0.570940	−	0.820992i	$-0.306579\pi$
$859$	33.4670	1.14188	0.570940	+	0.820992i	$0.306579\pi$
$860$	0	0
$861$	−39.8429	−1.35784
$862$	0	0
$863$	−8.18842	−0.278737	−0.139368	−	0.990241i	$-0.544507\pi$
$863$	−8.18842	−0.278737	−0.139368	+	0.990241i	$0.544507\pi$
$864$	0	0
$865$	−17.8926	−0.608365
$866$	0	0
$867$	−25.8135	−0.876672
$868$	0	0
$869$	10.4822	0.355584
$870$	0	0
$871$	31.5402	1.06870
$872$	0	0
$873$	−1.69808	−0.0574714
$874$	0	0
$875$	23.7307	0.802245
$876$	0	0
$877$	−5.91392	−0.199699	−0.0998495	−	0.995003i	$-0.531836\pi$
$877$	−5.91392	−0.199699	−0.0998495	+	0.995003i	$0.531836\pi$
$878$	0	0
$879$	48.7162	1.64316
$880$	0	0
$881$	−52.5604	−1.77080	−0.885402	−	0.464826i	$-0.846117\pi$
$881$	−52.5604	−1.77080	−0.885402	+	0.464826i	$0.846117\pi$
$882$	0	0
$883$	−18.9559	−0.637917	−0.318959	−	0.947769i	$-0.603333\pi$
$883$	−18.9559	−0.637917	−0.318959	+	0.947769i	$0.603333\pi$
$884$	0	0
$885$	−4.97118	−0.167104
$886$	0	0
$887$	−21.8842	−0.734799	−0.367399	−	0.930063i	$-0.619752\pi$
$887$	−21.8842	−0.734799	−0.367399	+	0.930063i	$0.619752\pi$
$888$	0	0
$889$	17.1200	0.574187
$890$	0	0
$891$	43.3878	1.45355
$892$	0	0
$893$	−2.69328	−0.0901271
$894$	0	0
$895$	−8.60293	−0.287564
$896$	0	0
$897$	−64.7713	−2.16265
$898$	0	0
$899$	−30.7212	−1.02461
$900$	0	0
$901$	1.25277	0.0417359
$902$	0	0
$903$	−38.9659	−1.29670
$904$	0	0
$905$	16.1184	0.535795
$906$	0	0
$907$	35.5215	1.17947	0.589735	−	0.807597i	$-0.299232\pi$
$907$	35.5215	1.17947	0.589735	+	0.807597i	$0.299232\pi$
$908$	0	0
$909$	1.14241	0.0378915
$910$	0	0
$911$	−16.4978	−0.546596	−0.273298	−	0.961929i	$-0.588114\pi$
$911$	−16.4978	−0.546596	−0.273298	+	0.961929i	$0.588114\pi$
$912$	0	0
$913$	43.5151	1.44014
$914$	0	0
$915$	3.34322	0.110523
$916$	0	0
$917$	5.92814	0.195764
$918$	0	0
$919$	−26.4940	−0.873956	−0.436978	−	0.899472i	$-0.643951\pi$
$919$	−26.4940	−0.873956	−0.436978	+	0.899472i	$0.643951\pi$
$920$	0	0
$921$	4.16800	0.137340
$922$	0	0
$923$	−28.2471	−0.929766
$924$	0	0
$925$	20.2450	0.665650
$926$	0	0
$927$	3.19385	0.104900
$928$	0	0
$929$	−49.4200	−1.62142	−0.810709	−	0.585449i	$-0.800918\pi$
$929$	−49.4200	−1.62142	−0.810709	+	0.585449i	$0.800918\pi$
$930$	0	0
$931$	−3.53345	−0.115804
$932$	0	0
$933$	−4.71450	−0.154346
$934$	0	0
$935$	5.06360	0.165597
$936$	0	0
$937$	−52.6725	−1.72073	−0.860367	−	0.509675i	$-0.829766\pi$
$937$	−52.6725	−1.72073	−0.860367	+	0.509675i	$0.829766\pi$
$938$	0	0
$939$	−55.7447	−1.81916
$940$	0	0
$941$	12.5292	0.408441	0.204221	−	0.978925i	$-0.434534\pi$
$941$	12.5292	0.408441	0.204221	+	0.978925i	$0.434534\pi$
$942$	0	0
$943$	−58.8552	−1.91659
$944$	0	0
$945$	13.5290	0.440099
$946$	0	0
$947$	40.5778	1.31860	0.659301	−	0.751879i	$-0.270852\pi$
$947$	40.5778	1.31860	0.659301	+	0.751879i	$0.270852\pi$
$948$	0	0
$949$	53.9192	1.75029
$950$	0	0
$951$	−44.9508	−1.45763
$952$	0	0
$953$	1.78729	0.0578960	0.0289480	−	0.999581i	$-0.490784\pi$
$953$	1.78729	0.0578960	0.0289480	+	0.999581i	$0.490784\pi$
$954$	0	0
$955$	0.827440	0.0267753
$956$	0	0
$957$	32.4645	1.04943
$958$	0	0
$959$	−47.2834	−1.52686
$960$	0	0
$961$	36.5821	1.18007
$962$	0	0
$963$	2.16348	0.0697170
$964$	0	0
$965$	8.74875	0.281632
$966$	0	0
$967$	−2.08006	−0.0668901	−0.0334450	−	0.999441i	$-0.510648\pi$
$967$	−2.08006	−0.0668901	−0.0334450	+	0.999441i	$0.510648\pi$
$968$	0	0
$969$	−2.09574	−0.0673247
$970$	0	0
$971$	19.3316	0.620379	0.310190	−	0.950675i	$-0.399607\pi$
$971$	19.3316	0.620379	0.310190	+	0.950675i	$0.399607\pi$
$972$	0	0
$973$	75.3306	2.41499
$974$	0	0
$975$	35.4877	1.13652
$976$	0	0
$977$	−33.4729	−1.07089	−0.535447	−	0.844569i	$-0.679857\pi$
$977$	−33.4729	−1.07089	−0.535447	+	0.844569i	$0.679857\pi$
$978$	0	0
$979$	74.3731	2.37697
$980$	0	0
$981$	2.92402	0.0933568
$982$	0	0
$983$	−46.4778	−1.48241	−0.741205	−	0.671279i	$-0.765745\pi$
$983$	−46.4778	−1.48241	−0.741205	+	0.671279i	$0.765745\pi$
$984$	0	0
$985$	8.18188	0.260696
$986$	0	0
$987$	14.6228	0.465449
$988$	0	0
$989$	−57.5596	−1.83029
$990$	0	0
$991$	25.7412	0.817696	0.408848	−	0.912602i	$-0.365931\pi$
$991$	25.7412	0.817696	0.408848	+	0.912602i	$0.365931\pi$
$992$	0	0
$993$	1.20388	0.0382039
$994$	0	0
$995$	4.95891	0.157208
$996$	0	0
$997$	39.3208	1.24530	0.622650	−	0.782500i	$-0.286056\pi$
$997$	39.3208	1.24530	0.622650	+	0.782500i	$0.286056\pi$
$998$	0	0
$999$	24.6747	0.780673

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.d.1.15	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.15	✓	19	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.a (trivial)

\(f(q)\)	\(=\)	\(q+1.67288 q^{3} -0.778334 q^{5} +3.24553 q^{7} -0.201474 q^{9} +O(q^{10})\)	\(q+1.67288 q^{3} -0.778334 q^{5} +3.24553 q^{7} -0.201474 q^{9} -5.19304 q^{11} -4.82763 q^{13} -1.30206 q^{15} +1.25277 q^{17} -1.00000 q^{19} +5.42938 q^{21} +8.02017 q^{23} -4.39420 q^{25} -5.35568 q^{27} -3.73699 q^{29} +8.22083 q^{31} -8.68733 q^{33} -2.52611 q^{35} -4.60720 q^{37} -8.07604 q^{39} -7.33839 q^{41} -7.17686 q^{43} +0.156814 q^{45} +2.69328 q^{47} +3.53345 q^{49} +2.09574 q^{51} +1.00000 q^{53} +4.04192 q^{55} -1.67288 q^{57} +3.81793 q^{59} -2.56764 q^{61} -0.653888 q^{63} +3.75751 q^{65} -6.53327 q^{67} +13.4168 q^{69} +5.85114 q^{71} -11.1689 q^{73} -7.35096 q^{75} -16.8542 q^{77} -2.01851 q^{79} -8.35499 q^{81} -8.37951 q^{83} -0.975075 q^{85} -6.25153 q^{87} -14.3217 q^{89} -15.6682 q^{91} +13.7525 q^{93} +0.778334 q^{95} +8.42831 q^{97} +1.04626 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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